\(C^{1}\) regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two. (English) Zbl 1170.35400
Summary: We prove \(C ^{1}\) regularity of \(c\)-convex weak Alexandrov solutions of a Monge-Ampère type equation in dimension two assuming only a bound from above on the Monge-Ampère measure. The Monge-Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely, non-negative cost-sectional curvature, a condition introduced in [X.-N. Ma et al., Arch. Ration. Mech. Anal. 177, No. 2, 151–183 (2005; Zbl 1072.49035)], that was shown in [G. Loeper, Acta Math. 202, No. 2, 241–283 (2009)] to be necessary for \(C ^{1}\) regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge-Ampère equation \(\det D ^{2} u = f\). Our result is in some sense optimal, both for the assumptions on the density (thanks to the regularity counterexamples of X.-J. Wang [ Proc. Am. Math. Soc. 123, No. 3, 841–845 (1995; Zbl 0822.35054)]) and for the assumptions on the cost-function (thanks to the results of Loeper [loc. cit.]).
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
References:
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